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A 60 -HOUR COURSE 




HARVEY D. WILLIAMS. 










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A 60-H0DR COURSE 


IN 


Kinematic Drawing 


BY 


' r 

Harvey D. Williams 


Assistant Professor of Mechanical Drawing in Sibley College 



Cornell University 


1 893 







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I 


Copyright, 1893, by 

HARVEY D. WILLIAMS. 



MADE IN 

THE COMPLETE ART-PRINTING WORKS 
OF 

THE MATTHEWS-NORTHRUP CO. 


BUFFALO,N.Y 


/ 



16089 






Introduction. 


1. Kinematics is the science of time and space or pure motion. 
It regards material bodies as having only the two attributes of 
position and form. A change of position is a displacement, and a 
change of form is a strain. Thus there are two kinds of motion, 
but it is only with displacement that we 
are at present concerned. 

2 . A free body, or a body capable 
of any displacement, has six degrees of 
freedom — three translations in direc¬ 
tions which are neither parallel nor in 
the same plane, and three rotations 
about axes which are neither parallel 
nor in the same plane. Such a body 
can take any position in any part of 
space. 

3. Corresponding to the six de¬ 
grees of freedom are six degrees of con¬ 
straint. For each degree of constraint 
that is applied to a body one degree of 
freedom is lost. Thus a body may 
have five degrees of freedom and one 
of constraint, or four of freedom and 
two of constraint, etc., as illustrated in 
Fig. 1. 

4. A body has one degree of free¬ 
dom and five of constraint when each 
point in it. that moves is free to de¬ 
scribe one line and only one. From 
our point of view this case is much 
•more important than the others, and 
the term constrained motion will be 
restricted to motion with one degree of 
freedom. Mechanism may be defined 
as the science of constrained motion ; 
i. e., motion of one degree of freedom. 

5. All motion about which we can 

know anything is relative. Hence Figl. 



3 








































there will be under consideration in any given case at least two 
bodies. When these two bodies move in contact with each other 
they constitute a kinematic pair. Each of the bodies is a kinematic 
link, and the surfaces which determine their relative motion are 
called elements. 

6 . A single pair of elements may allow more than one degree 
of freedom, but it is always possible to connect it with other pairs 
in such a way as to produce perfect constraint and so make it avail¬ 
able for use in mechanism. 

7. When the elements of a pair are of equal and constant 
curvature there may be surface contact between them during 
motion, otherwise there can be only line or point contact. Pairs 
are thus divided into two classes : primary or lower, those in which 
surface contact is possible ; and higher or secondary, those in which 
surface contact is impossible. 

8. The variety in the forms of elements of higher pairs is with¬ 
out limit ; since any two surfaces may touch at a point, and any 
two surfaces generated by the motion of the same line may have 
line contact. 

9. The number of surfaces of constant curvature is, however, 
limited to six, of which three have constant curvature in one direc¬ 
tion only and three have constant curvature in more than one direc¬ 
tion. Hence there will be six primary pairs, as in Fig. 2. 

10. One of the elements of a primary pair may be reduced to 
a number of lines or points without affecting the motion provided 
the other element is left intact. Thus three points, or a right line 
and a point, may be equivalent to a plane, and four points, or a 
circle and a point, may be equivalent to a sphere, etc. A round 
shaft will have the same motion whether fitted to a round or a 
square hole, but in the latter case there will be four lines of contact 
instead of a cylindrical surface. Primary pairs are frequently given 
line or point contact in this way for the sake of accuracy, as, for 
example, the V supports of a level or transit instrument. Chances 
of inaccuracy due to the presence of dust evidently decrease with 
the area of the contact surface. The avoidance of friction is 
another reason for making similar changes, as in knife-edge ful- 
crums and roller or ball bearings. 

11. By combining the primary pairs it is easy to get very com¬ 
plex motions. But these motions, however complex, are always 
capable of being expressed by mathematical equations. It follows, 


4 


example 


SURFACE 


NAME. 


MOTION 



Any prism or 

CYLINDER EXCEPT 
A RIGHT CIRCULAR 
CYLINDER . 


SLIDING 

PAIR 


TRANSLATION 
IN ONE 
DIRECTION 



Any surface of 

REVOLUTION EX¬ 
CEPT A SPHERE 
OR ARIGHT 
CIRCULAR CYLINDER 


Turning 

Pair 


ROTATION 
ABOUT A FIXED 
AXIS 



ANY HELICAL 
SURFACE OF 
UNIFORM 
PITCH 


Twisting 

PAIR 


ROTATION ABOUT 
AN AXIS AND 
TRANSLATION IN 
DIRECTION OF AXIS 
IN FIXED RATIO 
TO THE ROTATION. 



PLANE 



SPHERE 



RIGHT 

CIRCULAR 

CYLINDER 


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TRANSLATIONS 
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ROTATION 
ALL IN ONE PLANE. 


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THREE 

Rotations about 

AXES WHICH HAVE 
ONE POINT IN 
COMMON. 


TRANSLATION 

ROTATION 

OR 

TWIST. 


Fig. 2. 

therefore, that no such combination can be made to give a purely 


arbitrary motion. 

12 . Primary pairs have several structural advantages, which 
make them very important to the designer. Their forms are simple 
and easily produced by machinery. Their wear is slight, owing to 
the surface contact, and when worn they are easily adjusted without 
changing the character of the motion. It is customary to make 
the material of one element much harder than the other, in order to 
preserve the original form of the surface, for if both surfaces are 
allowed to wear they may gradually lose their surface contact or 





















































become transformed into another primary pair. Thus the 
sliding pair may become a turning pair. This has been known 
to occur with an engine slide valve, the originally flat valve seat of 
which became transformed into a surface of revolution of less than 
24" radius. 

13. By means of a higher pair any motion whatever may be 
produced. One element can always be a surface of constant curva¬ 
ture, and if the motion cannot be produced by primary pairs at 
least one element must be a surface of varying curvature. In a 
perfectly constrained higher pair there will be more than one con¬ 
tact, and each element will consist of two or more surfaces, which 
are related to each other in such a way that when certain ones are 
known the others may be deduced from them. Thus the acting 
faces of gear teeth are not wholly arbitrary, but half of them are 
deducible from the other half. 

14. An element of varying curvature will generally wear un¬ 
equally in its different parts, and the slightest wear, even when 
uniformly distributed, will change the character of the motion. 
Matters are still worse when the element consists of two or more 
irregular surfaces having a proper mathematical relation to each 
other. Hence, it is that when higher pairs are used in machinery 
they are arranged so as to give only one or two degrees of con¬ 
straint, and other constraint is obtained by connecting them with 
primary pairs. (§ 6.) 

15. Any plane or spheric motion can be produced by a pure 

rolling of two surfaces on each 
other. Thus the wear may be 
almost eliminated in many cases 
by substituting rolling for sliding 
contact. 

16. In a machine each group 
of pieces having one or more ele¬ 
ments, in the latter case without 
relative motion, is a link. Fig. 3 
shows a link having elements of 
two turning pairs. Fig. 4, a link 
having an element of a turning 
pair and an element of a sliding 
pair. Links may be very simple 
in form, as in Fig. 5, which con- 

6 























nects an element of a partially constrained turning pair with one of 
a partially constrained twisting pair, or Fig. 6, which connects 
elements of two sliding pairs, one partially and the other wholly 
constrained. A compound link is one containing more than two 
elements, as in Fig. 7. 

17. A number of links connected together by their paired ele¬ 
ments constitute a kinematic chain. (Fig. 8.) A 
kinematic chain in which the motion of each link 
in relation to every other link is constrained is a 
closed chain or linkage. (Fig. 9.) 

18. In designing machinery conditions are im¬ 

posed by kinematics wherever there is constrained 
motion. Frequently, one 
very simple condition is all 
that it imposes; for ex¬ 
ample, the rubbing surface 
of an eccentric strap must Fig. 7 . 

be a surface of revolution, since it is an element of 
a turning pair. In such cases the designer is con¬ 
cerned principally with questions of strength, dura¬ 
bility, facility of repair, cost of construction, etc. 
Such considerations are nearly always leading ones, 
and in comparatively few cases do the kinematic 
requirements become so important as to need 
special investigation by the designer. 

19. When kinematic requirements do present 
themselves, however, they are imperative, and 
there is no such thing as a factor of safety in 
dealing with 
them. 

Care 
b e 


Fig.9. 


Fig. 8. 





taken to dis¬ 
tinguish between requirements that are essentially 
kinematic and those that result from mere expedi¬ 
ency. A gear tooth may, within certain limits, be 
made of any length. There are good reasons for 
making it long and equally good reasons for mak¬ 
ing it short, and the ground on which the length is 
finally decided may be the farthest removed from 


7 













































kinematics, for it is simply given a length such that it is easily 
measured and easily described to the end that all who have any¬ 
thing to do with the dimension may avoid mistakes. With the 
curved face of the tooth, however, the case is different, for here a 
single requirement is imperative, and the shape is defined with 
mathematical exactness, and must be produced with all attainable 
accuracy. 

DRAWING INSTRUMENTS. 

21 . Accuracy is of the first importance. Hence due care must 
be taken in the selection and use of drawing instruments. The 
drawing board and T square are apt to warp, and a few moments 
spent, now and then, in putting them in proper condition will save 
much annoyance. The face of the drawing board and its left-hand 
edge must be plane surfaces at right angles to each other. The 
head of the T square must have a plane surface at right angles to 
the blade, and the edge of the blade must be straight. When not in 
use the T square should be suspended vertically or lie, blade down, 
on a flat surface. Rubber triangles will warp if left in the sun. 

22 . Use a hard pencil and sharpen it to a conical point. The 
chisel point is well enough where nothing but straight lines are to 
be drawn, but it is objectionable for drawing irregular curves, and 
utterly useless for laying off dimensions from a scale. The conical 
point must, of course, be kept sharp. 

23. One double-jointed compass with removable points is worth 
more than a whole assortment of compasses with rigid legs. This 
is very important, and fault is sure to be found with work in which 
the latter instruments are used. In general it may be said that a 
few carefully selected instruments, kept in good order, are much 
better than a large collection of the cheaper variety. The needle 
and divider points should be sharp enough so that the mere weight 
of the instrument is sufficient to hold them in place. The compass 
lead should be sharpened to a chisel point, care being taken that the 
edge is parallel to the axis of the compass joint. 

24. In using the compasses the legs should be bent so as to 
stand normal to the paper, and in general the instruments should be 
held in as nearly vertical planes as possible. 

25 . When a line is to be drawn through two points place the 
pencil point at one of them and work the straight edge against it 
as a centre until it touches the other point. 


8 


26. Right lines and circles can be bisected, trisected, etc., much 
more readily by trial than by geometrical constructions if the error of 
each trial be noted and the dividers opened or closed by the estimated 
fractional amount of the error. To divide into six parts, first bisect 
and then trisect each half, and similarly for any other number of 
divisions when the number can be resolved into factors. 

2 7 - When a point has been located indicate its position by two 
cross lines or by a minute circle around the point. Punching a hole 
through the paper or obscuring the point with a black smudge of a 
soft pencil is no indication of its position. 

28. The solution of problems involving the relative motion of 
two bodies can often be facilitated by the use of tracing paper. 
For this purpose a paper should be selected which will not warp 
from the moisture of the hands. Two or three fine sewing: needles 
inserted in wooden or sealing-wax handles will be found convenient 
for laying off dimensions from scales, and also for use as instanta¬ 
neous centers in connection with the tracing paper. 

IRREGULAR CURVES. 

29. Irregular curves, or curves other than the circle, will occur 
in nearly all of the following problems. An irregular curve may be 
determined by a series of points through which the curve passes, in 
which case the curve is said to be the locus of the point, or it may 
be determined by a series of lines, either straight or curved, to each 
of which the curve is tangent, in which case the curve is said to be 
the envelope of the lines. 

30. Curves determined in either of these ways are drawn by 
sweeps, the use of which requires special care. First, draw the 
curve free hand lightly with a soft pencil, and as accurately as pos¬ 
sible, in order that the eye may catch the general form. Then for 
each part of the curve select a portion of a sweep that has about 
the same curvature, noting at the same time the rate at which the 
curvature changes. By turning the sweep over, the curvature may 
be made to increase in either direction. Apply the edge of the 
sweep to the curve and by trial find a position where the coincidence 
is the closest, then, with the sweep to guide the pencil, draw that 
portion of the curve. The difficulty is in selecting the sweeps, and 
it is evidently of advantage to haye a large variety from which to 
choose. 


9 


31. There is an adjustable curved ruler, consisting of a steel 
tape and a lead bar, by means of which irregular curves can be eas¬ 
ily drawn. The ruler is simply bent to the required shape, where it 
remains by virtue of the ductility of the lead. It does very well 
where the curvature is only moderate. 

32. Let any curve A be cut from a piece of cardboard and 
suppose an inelastic cord stretched along the curved edge. If the 
cord be unwound, holding it taut, a point in it will describe a curve 
B , which is the involute of A, and A is the evolute of B. Since any 
point in the cord may describe the involute, it follows that for each 
evolute there are an infinite number of involutes. Since the straight 
part of the cord is at each instant the radius of curvature of the 
involute, and therefore normal to it, it follows that involutes of the 
same evolute are parallel curves. 

33- It sometimes happens that it is just as easy to determine the 
evolute of a required curve as the curve itself. When this is true 
there is afforded a simple method of drawing the required curve. 
For if a series of points be found in the evolute, each one may be 
made the centre of a circular arc, which will agree very closely 
with a portion of the required curve. It only remains to see that 
adjacent circular arcs always join at a common normal. 


GRAPHICAL CONSTRUCTIONS. 

34. In order that the graphical constructions may be easily fol¬ 
lowed the lines will, in many cases, be numbered 1, 2, 3, 4, etc., in 
the order in which they are drawn, letters being used to designate 
points of intersection, subdivision, etc. The following construc¬ 
tions are of a general character, and are applicable to curves 
whose mathematical properties -are unknown. Where exact meth¬ 
ods are known they should be given the preference. 

35. In solving problems in connection with gear teeth, it will 
frequently be necessary to lay off equal spaces along the arcs of 
different curves. On circles having the same radius, equal arcs 
can be spaced off with a pair of dividers, but when the radii are 

unequal an empirical method based on 
the principle illustrated in Fig. 10 is 
used. If O B is one fourth of O A a 
circular arc through A and with B as a 
centre will cut equal arcs from the sev¬ 
eral circles, so that arc O C = arc O D 



IO 





= O A = arc O E. The circles are tangent to each other and to 
the straight line at O. Arcs of equal length found in this way may 
be sub-divided by continued bisecting (§ 26), thus giving the series 
of spaces required. This method should not be applied to arcs of 
over 6o°, as, being empirical, it is not absolutely accurate. 

36 . The construction shown in Fig. 
11, deduced from the preceding, gives a 
means of developing a circular arc into a 
straight line. O C is the given arc. Pro¬ 
long its chord beyond O , making O B 
equal to one half of the chord O C. A 



Fig. 11 . 


circle through C, center at B, cuts from the tangent at O a distance 
O A , equal to the arc O C. 

37 - To find a tangent point on an irregular curve: Let the 
line 1 (Fig. 12) be tangent to the curve 2. To find the point of 
tangency, draw two or three chords (as 3, 4) parallel to the tangent, 
and through the extremities of each chord draw parallel lines 5, 6, 
7, 8, and on these lines measure off, from where each intersects the 

tangent, distances equal to the respec¬ 
tive chords ; thus, C D=A B=E E. A 
smooth curve (9) passing through the 
points so found will cut the given curve 
at the required tangent point. 

38. To draw a tangent through a 
given point on an irregular curve : Let 
1 (Fig. 13) be the given curve and A 
the point through which a tangent is to 
be drawn. Draw a circular arc (2) with 
center at A. On a secant (3) passing 
through A lay off C D=A B. The 
locus of D will be a curve (4) which 
cuts the arc 2 at E, a point in the required tangent. 

39. To draw the evolute of 
an irregular curve : Let 1 (Fig. 
14) be the given curve, 2 a tan¬ 
gent and A its tangent point. 
(§§ 37, 3 8 -) Draw the normal 3. 
Find by trial the center ( B ) of 
a circle that agrees with a rea¬ 
sonably small portion of the 
















given curve near A , say from A to C. 

Draw the normal C B and find by 
trial the center ( D) of an arc (C E) 
that agrees with another portion of 
the curve, and so on to the end of the 
curve. The broken line, B D F H, 
is the evolute required. (§ 33.) 

40. To lay off equal spaces along 
the arc of an irregular curve : Draw 
a fine straight line (Fig. 15) on a strip 
of tracing paper (§ 28) and divide it 
into parts at A B C B>, etc., equal 
to those required on the curve. Lay off one quarter of a part at 
An, Bb, Cc , etc. Place the tracing over the given curve so that the 
point A is on the curve. Insert a needle (§ 28) at A and turn the 
tracing about it as a center until the straight line is tangent to the 
curve at A. Insert another needle at a, and, removing the first 

needle, turn the tracing until 



A 

—o—o- 


B 


a 


C 

- 0 — 0 - 


B 


b C 

Pig*. 15. 


point B is on the curve. Insert 
a needle at B, remove the one at 
a and turn again until B is the 


tangent point. Then place a needle at b and remove the one at B 
and rotate until C is on the curve, etc. This method is based on the 
one described in § 35, and is subject to the same limitations. 

41. To lay off equal spaces along circular arcs or along irregu¬ 
lar curves when their evolutes are known : Draw on tracing paper 
three lines as in Fig. 16, making A B equal to 
one of the required spaces. C A and B D are 
at right angles to A B , and each is longer than 
the longest radius of curvature of the given 
curve. Lay off A a = B b = one-fourth of A B. 

Place the tracing over the given curve so that 
the point A is on the curve. Insert a needle at 
A and turn the tracing about it as a center until 
AC is tangent to the evolute, or, in case of a 
circle, until A C passes through its center. 

Insert a needle at a, remove the one at A, and 
turn the tracing until B is on the curve. In¬ 
sert a needle at B, remove the one at a and turn 
until B D touches the evolute. 


A 


a 


Fig*. 16. 


B 


12 










42. lo draw a parallel to a given curve: Let i (Fig. 17) be 
the given curve. With center on 1 and radius equal to the distance 

between the curves draw a circu¬ 
lar arc, 2. The required curve 
will be the envelope of the arc 2. 
By drawing a sufficient number of 
circles the curve may be deter¬ 
mined to any desired degree of 
accuracy. 

43. If the given curve is on tracing paper, to draw a parallel 
curve : Lay the tracing over two parallel lines, which are the 
required distance apart. Place the curve tangent to one line and 
trace the other. The required curve will be the envelope of the 
second line. 

44. To draw a parallel to a given curve when its evolute is 
known : Find a series of centers on the evolute by means of which 
the given curve can be drawn with circular arcs. With the same 
centers and with similar arcs draw the required curve (§ 33). 

45- To find the circumference of a given circle : Draw the 

diameter A B 
(Fig. 18) and 
from one ex¬ 
tremity lay off 
three diame¬ 
ters on the 
tangent C B. 
Draw the ra¬ 
dius D E making 30° with A B and draw E F parallel to C B. C F 
will be the circumference required. This is equivalent to 7t = 

mb + 

46. If two plane curves roll on each other in the same plane a 
point attached to one will describe a roulette in relation to the 
other. The curve to which the point is attached is a generating 
curve and the other is a directing curve. 

If the directing curve is a straight line and the generating 
curve is a circle the roulette will be a cycloid or a trochoid accord¬ 
ingly as the describing point is on or off the circumference. If 
both the directing and generating curves are circles, the roulette 
will be an epicycloid, or hypocycloid, or epitrochoid, or hypo- 
trochoid, according to the conditions as shown in Fig. 19. 




1 


o 












Fig*. 19. 


47. To draw a rou¬ 
lette : Let 1 (Fig. 20) be 
the directing curve, 2 the 
generating curve and A 
the describing point, one 
or the other of the curves 
being on tracing paper. 
On a strip of tracing 
paper, or, what is much 
better, a strip of trans¬ 
parent sheet gelatin, draw 
a straight line, 3, and lay 
off on it a distance C D 
of such length that when 
applied as a chord, E F, 
to the sharpest part of 
either curve the greatest 
distance, G JZ, from the chord to the curve shall not be greater 
than one eighth of the length of the chord. Lay off C c and D d, 
each equal to one fourth of C D. Place the two curves tangent to 
each other and to the straight line at the point C , the tracing of the 
straight line being uppermost. This is the position shown in Fig. 
20. Prick the position of 
the describing point A and 
insert a needle at C. Turn 
both tracings about C as a 

center until the curves in¬ 
tersect at D. Insert a 
needle at Z), remove the 
one at C and turn the 
tracing until the curves are tangent at 
D. This gives another position of the 
describing point A. By turning the 
line C D through half a circumference about 1) as 
a center the conditions will again be as shown in 
the figure, and the operation can be repeated. 

48. To draw a roulette when the generating 
and directing curves are circles or curves whose 
evolutes are known : Draw the lines 4 and 5, Fig. 21, at right 
angles to C D. If two curves are tangent to each other and to CD 


4 


D 


Fig.21. 



Fig. 20. 



























at C their evolutes will be tangent to 4, or in case of circles, their 
centers will be on 4. 1 hus the curves can be placed accurately 

tangent to each other. In other respects the method will be as 
described in § 47. I 


Si 


0 

b 



m 

CAMS. 

49. A cam and its follower (Fig. 22) are links 
connected by higher pairing for the purpose of 
getting an arbitrary motion (§ 13). Certain objec¬ 
tionable features restrict their use to cases where 
the motion is difficult to produce accurately or even 
approximately by primary pairing (§ 11). The 
principal objection is that, owing to the smallness 
of the area of contact, a moderate force may pro¬ 
duce a very severe pressure per square inch be- Fig 1 . 22. 
tween the rubbing surfaces, and the result will be rapid wear (g 14). 
Abrasion may be prevented by making the cam very large, thus 
slightly increasing the area of contact, or by inserting pieces of 
hardened steel at points where the wear is most severe (A, Fig. 22), 
or by substituting rolling for sliding contact (B, Fig. 25) (§ 15). 

50. A circular cam 
with a flat follower, as in 
Fig. 23, will give a har¬ 
monic motion. The use 
of such a cam for that 
purpose, however, would 
most likely show bad 
judgment, seeing that primary pairs arranged 
as in Fig. 24 will give the same motion, and are 
at the same time more compact and capable of 
transmitting many times as much force without cutting. 

51 . Let A, B, C, and B, Fig. 26, rotate with equal uniform angu¬ 
lar velocity in the direction of the arrows. Let their forms be such 
that the followers E , F, G, II are caused to move vertically with 
uniform velocity. Suppose the followers move through a distance 
a per revolution. Then A will be a circle whose circumference is 
a. B will be an involute of A. C will be an Archimedean spiral, 
whose polar equation is r= a n , where r is the radius vector and 
n its angular position in circumferences. D will be a parallel of 
the Archimedean spiral C, the distance between the parallel curves 



Fig*. 24. 


Fig. 23. 


15 

































being equal to the radius of the roller II. The relation of the 
curves to each other will be rendered more apparent by noticing 
the manner in which they are generated. Let A , Fig 26, be a circle 
of the same diameter as A , Fig. 25. On the right angle E b K lay 
off be — radius of A and about c as a center draw the circle d of 
the same diameter as H in Fig. 25. Suppose the right angle is 
rolled on A by the side b E without sliding, then the locus of b will 
be the involute of a circle, the locus of c will be an Archimedean 
spiral, and the envelope of d will be a parallel of the Archimedean 
spiral, the curves being equal in every respect to those in Fig. 25. 

52 . If the vertical motions of E, E, G , H y Fig. 25, are uni¬ 
formly accelerated so that at each instant the space traveled per 
revolution is varying by an amount a per revolution, then A will be 
an involute of a circle. The diameter of the circle being B 
will be an involute of A. C will be the same as the relative path of 
a point c vertically over the center of rotation, and D will be a par- 
allel of C. The polar equation of C in this case is r — an , where n 

is measured in circumferences. 

53. Whatever be the form of A , Fig. 25, 
then to produce the same motion, B will be an 
involute of A. C will have the form of the path 
of c in relation to B } and D will be a parallel of 
C. So that if any one of the forms A, B, C, D 
is known the others may be easily determined. 
And if the law of the motion is known, C may 
be determined by its polar equation. 

16 


in 






Fig',26. 























































54* I h e considerations which determine one’s choice of an 
arbitrary motion will differ widely in different cases, and they lie for 
the most part outside of our present subject. It may be well, how¬ 
ever, to mention one principle which has a somewhat general appli¬ 
cation in those cases where the work done, rather than the motion, 
is of primary importance. 

A part of a machine will convey energy at a rate proportional 
to the force which it transmits. The strength of the part being 
constant it follows that to convey the maximum of energy the force 
should be constant. Thus, it may be desirable to design a machine 
in such a way that a certain part, the driving part for instance, shall 
be subjected to a constant stress. 

Suppose a part A must of necessity act against a varying resist¬ 
ance, the resistance at each instant being a function of its position. 

A is driven by a series of links, among which is the link B. It 
is desired that B shall be subjected to uniform stress. Assume for 
the time being that B moves uniformly in the direction of the stress 
and then give A an arbitrary motion such that at each instant its 
velocity in the direction of its resistance is inversely proportional 
to the resistance. The velocity ratio being determined in this way, 
the stress on B will be uniform whether its motion is uniform or not. 

In general, we may say that the rate of doing work should be 
constant. For example, if work is done against a constant resistance, 
as a friction or the weight of a body when lifted against gravity, 
then the velocity in the direction of the resistance should be con¬ 
stant. If the work consists in overcoming the inertia of a heavy 
mass then the motion should be uniformly accelerated. A cam is 
often used to close a clamp or vice. If the piece to be clamped 
is inelastic a spring resistance is inserted between the follower and 
the clamp. Here the motion should vary inversely as the space. 
It is easy to carry this principle too far. Thus various methods 
have been contrived to do away with the crank and connecting rod 
of the steam engine, the object being to equalize the tortional 
moment on the crank shaft. The object is desirable enough but 
there appears to be no easy way of accomplishing it. 

55. Prob. 1. A cam like that shown in Fig. 22 rotates with 
uniform angular velocity. The follower is held in contact by its 
weight. The follower must rise with uniform velocity during two 
thirds of each revolution. Determine the form of the cam that may 
be run at the greatest number of revolutions without leaving contact 


with the follower Let the follower move 2^2" and be 1%" from the 
center of the cam shaft at its lowest position. 

56. The rising part of the cam will in this case be an involute of 
a circle (§ 51) and the falling part will be an involute of an involute 
of a circle (§52). For, since the follower drops by its own weight, 
that form of cam can be run at the greatest speed which allows it 
to drop with the uniformly accelerated motion of a free falling body. 

57 - T he preceding paragraph will suggest one method of solving 

the problem, but the following solu¬ 
tion is simpler and of more general 
application. Let A , Fig. 28, be at 
the center of a piece of tracing paper 
(8" x 8") (§ 28). Through A draw 
twenty-four radial lines 15 0 apart, 
and number them o, 1, 2, 3, etc., in 
such order that the rotation of the 
cam as indicated by the arrow will 
bring them successively to the same 
position. These lines can be most 
easily drawn with the 30° 



and 45 


Fig*. 28 . 

triangles. For by 
placing the triangles 
separately and to¬ 
gether in various 
positions against the 
T square, all of the 
angles can be pro¬ 
duced. 

58. Let A , Fig. 

27, be the center of 
the cam. The great¬ 
est radius of the cam 
will be 4", so that A 

should be about 4 }^" Fig*. 27. 

from the margin of the paper, o will be the lowest position of the 
follower and 16 its highest position. The dimensions are changed 
from those given in the problem to allow for a practical difficulty 
which will appear later. Divide the space between o and 16 into six¬ 
teen equal parts and make the divisions o, 1, 2, 3, etc., beginning at 
the bottom. Lay a scale of equal parts diagonally across the lines 



.c-> 


18 







































o, 1 6 , at such an angle that there will be sixty-four divisions between 
the points of intersection B and C. Mark the position of the first 
division, counting from B, also the fourth, ninth, sixteenth, twenty- 
fifth, thirty-sixth, etc. Through the points thus laid off from the 
scale draw horizontal lines and number them 16, 17, 18, etc., begin¬ 
ning at the top (§ 69). 

59. Lay the tracing, Fig. 28, over Fig. 27, so that the points A 
coincide, and, inserting a needle at A , turn the tracing about it as a 
center until o on the tracing coincides with D. Trace the line o 
corresponding to the lowest position of the follower. Turn the 
tracing through one division so that 1 coincides with D and trace 
the corresponding position of the follower, and so continue, alter¬ 
nately rotating and tracing through the twenty-four divisions. The 
envelope of the lines traced will be the required outline of the cam. 

60. The outline will include a flat place, which is objectionable 


in practice, be¬ 


cause the fol¬ 
lower will strike 
it with a blow 
at each revolu¬ 
tion. With A 
(Fig. 29) as a 
centre and a ra- 



dius of i/4", 
draw the circle 
E and substi¬ 
tute for the flat 
place a circular 
arc which is tan¬ 
gent to E at D 
and to the cam 
outline at B and 
C. The lowest 



position of the follower will then be 1 1 / 2 " from the center of the 
shaft, as the problem requires. The drawing may be completed 
according to the dimensions in Fig. 29. The key-seat may be 
placed in any position except near B>, where it would weaken what 
is already the weakest part. 

6l. Prob. 2. Let the cam of the alligator shear, Fig. 30, rotate 
as indicated by the arrow, with uniform angular velocity. The fol- 


19 












































































lower remains in its lowest position during one sixth of a revolution 
and then rises, turning about the points, during seven twelfths of a 
revolution, and falls during the remaining quarter. Let the fol¬ 
lower rise with such angular velocity that the point of intersection, 
B, of its lower edge with the line 3 travels along the line 3 with 
uniform velocity, and let the fall be such that the point of inter¬ 
section, C, travels along the line 4 with uniformly accelerated veloc¬ 
ity. Make the greatest radius of the cam 4" and the least radius 
1 y 2 ". 


62. Assuming the upward motion to be along the line 3 as above 
has the effect of increasing the angular velocity of the follower at the 
latter part of its stroke where the resistance is least (§ 54), and assum¬ 
ing the downward motion on the line 4 has the effect of increasing the 
angular acceleration at the latter part of the motion where 
the weight of the follower acts at a greater leverage. 

63 - On a piece of tracing paper (8" x 8") 
draw the twenty-four radial lines and number 
them as in Fig. 28 (§ 57). Assume the 

position F t Fig. 31, of the 
center of the cam shaft, 
taking care that there is 
room enough to complete 
the drawing to the dimen¬ 
sions given in Fig. 30. 



Fig*. 31. 


20 




















































Draw the circular arcs i, 2, with radii respectively equal to the least 
and greatest radius of the cam, and draw 3, 4, as in Fig. 30. Locate 
the center A about which the follower turns and draw the arc 5 
with a radius equal to the perpendicular distance from A to the flat 
surface of the follower. 6 and 7, tangent to 5 and to 1 and 2, will 
be the extreme positions of the follower. 

64. Since the revolution of the cam is divided into twenty-four 
intervals, fourteen of them, or seven twelfths of the whole, will be 
the period during which the follower rises, hence, divide BD into 
fourteen equal parts and number the divisions o, 1, 2, 3, etc., begin¬ 
ning with o at the point B. The fall lasts one quarter of a revolu¬ 
tion, or six of the twenty-four intervals, therefore, divide EC into 
six parts, and since the motion is uniformly accelerated, these parts 
will be proportional to the numbers 1, 3, 5, 7, 9, n, increasing 
downwards (§ 69). To divide the line in this manner draw two 
parallel lines through its extremities, and across the parallel lines 
lay a scale of equal parts at such an angle that thirty-six of the 
parts fall between the two lines. Mark the required divisions along 
the edge of the scale and project them on the line EC by lines 
parallel to those already drawn. Number the divisions on EC 14, 
15, 16, etc., beginning at E. 

65. Lay the tracing over the drawing so that the focus of the 
radial lines coincides with E. Turn the tracing about E until its 
line o coincides with 4 and trace the line 6. Turn the tracing 
through one interval, making its line 1 coincide with 4, and trace a 
line tangent to 5 through the point 1. Turn the tracing another 
interval and trace a line tangent to 5 through the point 2, and so 
continue through the twenty intervals. For the four remaining 
intervals, during which the follower does not move, trace the 
line 6. 

66. The envelope of the lines traced will include two flat places 
(§ 60), for each of which substitute a circular arc of 8" radius. 
Complete the drawing to the dimensions given in Fig. 30. 

67. Prob. 3. Fig. 32 shows a type of cam that one sees in the 
engine-valve gears of American side-wheel steamers. The crank 
A turns about the center B with uniform angular velocity. The 
connecting rod C is so long that its angularity may be neglected. 
The cam is rigidly connected to the rocker-arm E>, which swings to 
and fro through an angle of 90°. M and N are the extreme posi¬ 
tions of the follower. During half a revolution of the crank, while 


21 



remains stationary in its lowest position, M. During a quarter of a 
revolution, while D swings from F to H, the follower rises to the 
position JV, and during the remaining quarter it falls back to M. 

68. It is desired to make the motion of the follower uniform. 
But a reciprocating motion of uniform velocity can only be approx¬ 
imated, and the closer the approximation the greater will be the 
stress and shock of starting and stopping, it being physically impos¬ 
sible for any mass to pass from a state of rest to a state of motion 
without passing through an intermediate state of acceleration. 
There is also the difficulty that the cam moves with infinite slowness 
at the end of its stroke, and hence would need to be of infinite 
length to give the follower a finite velocity. We will, therefore, 
make the following assumption in regard to the motion of the fol¬ 
lower : If the period during which it rises be divided into twelve 
equal intervals then the motion shall be uniformly accelerated 
during the first two, remain constant during the next eight, and be 
uniformly retarded during the last two. 

69. If a body starting from rest is uniformly accelerated in the 
direction of its motion, and the motion is divided into equal inter¬ 
vals of time, then the initial velocity will be o, and if the space cov- 
ered during the first interval is 1 the velocity in space per interval 
at the end of the first interval will be 2, the space covered during 


22 






















































the second interval will be 3, and at its end the velocity will be 4, 
the space for the next interval will be 5, and its final velocity 6, etc. 
The spaces covered during successive intervals being represented 
by the numbers 1, 3, 5, 7, etc., the total space for n intervals will 
equal n *. If a body is uniformly accelerated from rest during two 
intervals, and after that the motion is uniform, the spaces covered 
during successive intervals may be represented by the numbers 1, 3, 

4, 4, 4, etc. If the period of acceleration is divided into three 
intervals the numbers will be 1, 3, 5, 6, 6, 6, etc. If into four, 1, 3, 

5, 7, 8, 8, 8, etc. 

70. Locate the point E , Fig. 33, of the center of the rock-shaft, 
and draw M and N as in Fig. 32. Divide the space between M 

and N into twelve parts proportional to 
- t A1 10 the numbers 1, 3, 4, 4, . . 4, 3, 1, and num- 

- y 8 ber the divisions as in Fig. 33. On a 

__ c 

—5 ^ piece of tracing paper (8" x 14") locate 

. ■ ■ ?2 the point E , Fig. 34, about 4" from the 

left end and 2" from the upper edge, and 
draw Eo in direction at right angles to 
the length of the paper. With center at E 
and radius of 3^2" draw the arc O of 90° 
to represent one quarter of a revolution of 
the crank A. Draw T parallel to Eo and 
R at 45 0 and through the point of inter¬ 
section, 12, draw the arc P. Divide O into twelve equal parts and 
project the points of division on P by lines parallel to Eo- 
Number the divisions as in the figure 
and through each one draw a radial line 
to the center E. The radial lines will 
represent successive angular positions of 
the cam corresponding to the twelve 
positions of the follower in Fig. 33. 

Lay the tracing over Fig. 33 so that the 
points E coincide and Eo coincide 
with EF. Trace the line M and turn 
the tracing about E so that E 1 coin- 
cideSt with EF and trace the position 
of the follower marked 1, and so con¬ 
tinue till all the positions of the follower are traced. The cam 
may be cut off at a point determined by the arc S, Fig. 32, whose 



Fig.33 

F 


23 























center is at £, and the lower edge of the cam should be drawn with 
a curve that gives it a gradual taper. 

71 . In Fig. 35 suppose the sphere A is held in contact with the 
horizontal plane by an active force such as gravity, and suppose 

the sphere B touches two par¬ 
allel planes. In both cases 
there is one degree of con¬ 
straint (§ 3), but they differ in 
that A is constrained only so 
long as a force keeps the bodies 
in contact, while B is constrained by virtue of its form, or, rather, by 
a resistance to a change of form. Resistance to a change of form 
we may call a passive force since it acts only when acted upon. In 
the one case, constraint is inseparable from the idea of force, 
while in the other the idea of force, as well as force itself, may be 
eliminated. 

72 . A pair of elements having five degrees of constraint is said 
to be closed. If the constraint is wholly due to the forms of the 
elements, the pair is form closed. If a force is necessary to the 
constraint, the pair is force closed. Cams are commonly spoken 




Fig-. 36. 


24 


8 


















































































































of as being covered or uncovered according as the pair is form or 
force closed. For example, the cams that have been considered 
thus far are uncovered. 

73. If a machine contains a link that is driven through a force 
closed pair, and the motion of the link is at any time accelerated 
in the direction of the closing force, then there is a limit to the 
speed at which the machine can be run depending on the magni¬ 
tude of the closing force. For high speeds, or speeds which are 
indeterminate, such links should always be driven through pairs that 
are form closed. 

74 - Prob. 4. The covered cam, Fig. 36, rotates with uniform 
angular velocity. The follower moves 3" with a harmonic motion 
of three double vibrations per revolution. The diameter of the 
rollers is 2" and the distance between their centers is 8". Deter¬ 
mine the form of the cam. 

75 . When the center of one roller is at its least distance from 

the center of the cam the other will be at its greatest distance- 
The sum of the two distances will be 8 " and their difference will be 
the travel = 3". Hence, the least 
distance will be 2%". Assume 
E, Fig. 37, as the center of the 
cam-shaft, and on the center line 
of the follower E, A locate the 
points o, 12, which are the ex- Fig. 37 . 

trerne positions of the center of one of the rollers. Draw the semi¬ 
circle 3' 7 in diameter with center at 6, and divide its arc into twelve 
equal parts. Project the divisions on the line EA and number 
them as in the figure. 

76. Draw on tracing paper the lines Eo and E 12, Fig. 38, at 
an angle of 6o°, and divide the angle into twelve equal parts by 

the lines E 1, E 2, etc. Lay the tracing 
over Fig. 37 so that the points E coin¬ 
cide and Eo coincides with EA. On 
the tracing draw a circle of 1" radius 
and center at o, Fig. 37. Turn the 
tracing so that Ei coincides with EA 
and draw another circle of the same 
radius and center at 1, Fig. 37, and so 
continue. The envelope of the several 
circles will be a curve which, repeated 



12. 



25 



















six times, is the outline of the cam. Three of the repetitions will 
be with the tracing turned face down. 

77. In the last problem we assumed certain dimensions and a 
harmonic motion for the follower. Those assumptions could have 
been anything else, subject to the following limitations : The roll¬ 
ers must be of the same diameter. Their centers and the center of 
the cam shaft must be in the same straight line, and only half of 
the motion can be arbitrary. If the motion of the follower during 
half a revolution of the cam is assumed, then its motion during the 
remaining half must be a repetition of the first half, in the same 
order, but in the opposite direction. Thus, if the motion is accel¬ 
erated at a certain rate from left to right during the first half, then 
it must be accelerated at the same rate from right to left during the 
second half. 

78. Covered cams that are subject to none of the above limita¬ 
tions can be made as illustrated in Fig. 39. Each pair has two con¬ 
tacts and two irregular surfaces. One irregular surface is always 



Fig*. 39. 


deducible from the other (§ 13). Thus in A and B the outlines 
are simply parallel curves. In B , by making the pin tapering and 
the groove of trapezoidal section, the lost motion resulting from 
wear can easily be taken up. 

79. Prob. 5. Fig. 40 represents an open and cross belt revers¬ 
ing gear. The driving shaft A has a single wide pulley and the 
driven shaft B has two loose pulleys and two fixed pulleys. The 
loose pulleys are placed on the outside, because their bearings must 

26 

































be oiled. By shifting the open belt the shafts will run in the 
same direction, and by shifting the cross belt they will run in 

opposite directions. 

Such a device is used for driving metal 
planing machines, and to get a quick re¬ 
turn motion the pulleys on the shafts are 
made of two sizes. During the forward or 
cutting stroke of the planer the shaft B is 
driven through a large pulley which gives 
a slow speed, and during the return stroke, 
where no useful work is done, the shaft is 
driven through a small pulley. 

8o. 1 here is needed a device, to be 

placed near the shaft B , for shifting both 
belts. 

It must operate on the belts at the 
points C, D , where they approach the 
pulleys. 

One belt must be shifted at a time, 
since both belts must never be on the 
fixed pulleys at the same time. 

The device must be operated by a 
single motion of a link which can be 
driven by the planer itself, thus making 
the whole operation automatic. 



Fig.40. 


An adjustment must be provided whereby the necessary motion 
of this driving link can be varied. For, since the first belt to be 
shifted will always be the one that, for the time being, is driving 
the machine, it follows that the momentum of the machine must be 
depended upon to shift the second belt. The adjustment is to pro¬ 
vide against the uncertainty as to how far the momentum will carry 
the machine after the first belt is shifted. 

8l. Fig. 41 represents a belt shifter that answers to the above 
conditions. It is a covered cam with two followers. The cam turns 
about the point A and the followers turn about the points B, C. 
When the cam is in its middle position, as represented in the figure, 
the portion of the slot which is between the two rollers is irregular 
in form and is the only portion that moves the followers. The 
rest of the slot is simply made up of circular arcs, with A as a center, 
and serves to hold either follower in a fixed position while the other 


2/ 




















I 



D 


Fig. 41. 

one is moving. Thus if the rod D is thrown to the right the lever 
E is moved and F is held fixed. Throwing the rod to the left 
moves F, and F remains fixed. 

The center of the belt hole in the lever F is placed i" from the 
edge of the pulley to allow for the angularity of the cross belt. 

82. Locate the points A, B , C, Fig. 42, by the dimensions given 
in Fig. 41. Draw the circular arcs 1, 2, that are described by the 
centers of the rollers, and locate the extreme positions 3, 4, of the 



* 



































































































































































center line of the shifter Z’. Make the angles 5,6 and 7,8 equal to 
the angle 3,4. Then the points of intersection, G, H, K, Z, will 
be the extreme positions of the centers of the rollers. 

83. I he point A is so placed that the line 9 passes through the 
points GH ; and the line 10 passes through the points K , Z, a 
condition that is unessential, but it renders the motions of the 
shifters as symmetrical as they can be made. By symmetrical 
motions we mean that the two shifters move alike, and each one 
moves the same in either direction. Perfect symmetry is impossible, 
because the arcs 1, 2 are curved in opposite directions. 

84. Divide the angle 9, 10 into six equal parts by radial lines, 
and divide the distance GH into six parts proportional to the num¬ 
bers 1, 3, 5, 5, 3, 1. With the center at A draw circular arcs through 
the points of division. The intersections of the circular arcs and 
the radial lines will determine a curve, GL, which is the path of the 
centers of the rollers in relation to the cam. 

The rollers are 1" in diameter. Hence, the outline of the cam slot 
will be the envelope of 1" circles whose centers are on the curve GL 

85 . The total angular motion of the cam will be twice the 
angle 9,10. The total angular length of the slot, measured from 
center to center of the terminal semicircles, should be 67^4°, or a 
trifle more than three times the angle 9,10. This gives a slight 
clearance between the roller and the end of the slot at each of the 
extreme positions of the cam. 

The stud on which turns the driving link D can be fixed at any 
point in the radial slot. Thus the distance that D must move can 
be adjusted. 

Finish the drawing to the dimensions given in Fig. 41. Parts 
that are not figured may be measured by the attached scale. 

86. In all the preceding problems the follower moves in a 
plane at right angles to the axis about which the cam rotates. 
When the follower must move in a direction parallel with the axis 
of the cam shaft a crown cam, Fig. 43, is used. 

The two forms occur with about equal frequency in prac¬ 
tice, although the latter, unless formed on a milling machine, is 
more difficult to make, on account of having a warped surface. 
The crown cam is also less easily defined in a drawing. 

87. Prob. 6. Design a covered crown cam, Fig. 43, that gives 
one double harmonic vibration to the follower for each revolution 
of the cam. Let the amplitude of the vibration be 1 ". 

29 * 


88. The form of the slot is most easily represented on the 
developed cylindrical surface of the cam. The workman can then 
wrap the development around the cylinder, and with a prick punch 
mark any number of points in the curve. 

Let the lower face of the cam be finished, so as to give a con¬ 
venient reference line for locating the curve. Draw AB equal to 
the circumference of the cam (§ 45), and draw the circle 2 of a diam¬ 
eter equal to the amplitude of the vibration. Divide the circumfer¬ 
ence of the circle 2 and the line AB into the same number of equal 
parts. The intersections of the horizontal and vertical lines 



ig.43 


through the points of division will give a sine curve 3, which is the 
development of the center line of the slot. The developed edges 
of the slot will be the parallel curves 4 and 5 (§ 42). 

89. In this particular case the center line of the slot is an 
ellipse in space, and by turning the cam to a certain position, that 
shown in the figure, the ellipse will be projected in a straight line, 
and the edges of the slot will be projected in lines that are very 
nearly straight. 

90. A right section across the slot has the form of a symmetri¬ 
cal trapezoid, and the element of the follower is a conical surface, 
to provide for taking up any lost motion that may result from wear. 



















































































































The adjustment is made by shifting the cone in the direction of its 
axis. 

I he vertex of the cone should be in the axis of the cam. 
Although it is desirable, for the avoidance of friction, that the cone 
should be as sharp as possible ; if it be made sharper than the 
above there may occur a condition when it will result in a reduc¬ 
tion of the wearing surface. Such a condition occurs when the 
length of the least radius of curvature of the center line 3 ap¬ 
proaches the half width of the slot. Under these conditions an 
edge of the bottom of the slot may be a curve which crosses itself. 
The double cusp, formed by the curve crossing itself, can have no 
real existence in the cam ; hence the reduction of the wearing sur¬ 
face. 

91 . If the direction of the desired motion is neither normal nor 
parallel to the axis of the driving shaft, the motion can still be ob¬ 
tained by a single cam ; though, in practice, it is generally simpler 
to get the motion by a combination of two or more cams, as we are 
thereby enabled to resolve the motion into components, and con¬ 
sider each component separately. A similar method can be used 
when the motion is defined by complex curves in either two dimen¬ 
sional or three dimensional space. 

92. The following illustration will make evident the possibility 
of getting any arbitrary motion by means of two elements. Let 
one element consist of three equal spheres rigidly connected to¬ 
gether, and let the other element consist of two tubes and two par¬ 
allel surfaces, also rigidly connected together, the inside diameter 
of the tubes and the distance between the parallel surfaces being 
each equal to the diameter of the spheres. With a sphere in each 
tube, and the remaining sphere between the parallel surfaces, we 
have a pair of elements with five degrees of constraint, for each of 
the tubes offers two degrees of constraint, and the parallel surfaces 
offer one degree. The axes of the tubes can be any two curves in 
space, and the parallel surfaces can be of any form ; hence the 
motion may be of the most general character. A third tube may 
be substituted for the parallel surfaces ; but the curve of its axis 
will be partly determined by the axes of the other two. Observe 
that with three tubes there will be line contact between the ele¬ 
ments. 

It follows, therefore, that any arbitrary ' motion may be con¬ 
strained by two elements having three lines of contact. 


31 


93. By introduc¬ 
ing force closure, the 
elements may be 
greatly simplified, as 
in Fig. 44, which 
represents a form of 
cam that can be made 
to give an arbitrary 
motion of the most 
general character. 


GEARING. 

94. A pair of gears are two bodies which are capable of driv¬ 
ing each other by means of projections from the one engaging with 
recesses in the other. Leaving out of consideration, for the present, 
a few forms which admit of only point contact, we will consider the 
general theory of those which admit of line contact. 

95- When, as with the line, there is more than one point of 
contact, a definite relation must connect the velocities of the points, 
lest a fast-moving point push apart the contact at a slow-moving 
point. This relation does not concern the actual motions of the 
points, but only those components of the motions which are normal 
to the surfaces in which the points lie, since motions that are tan¬ 
gent to the surfaces at the points of contact can have no effect in 
separating them. Consider, now, that the projections and recesses 
are reduced to infinitesimal dimensions. What was before a tan¬ 
gent surface at the point of contact, becomes now a tangent line, 
and any tangential motion that does not separate the contacts must 
take place along this line. The only lines that can slide on each 
other and remain in contact throughout their length are lines of 
equal and constant curvature, and of these there are only three 
pairs : the straight line sliding on a straight line, the circular 
arc on a circular arc of equal radius, and the helix on an equal 
helix. 

But in a pair of gears we have a relative motion besides the 
sliding, else the one could not drive the other. Of the three pairs 
cited above, only one admits of a relative motion other than the 
sliding, and that is the pair of straight lines, for a rigid straight line 



32 


























































































can revolve about itself and at the same time slide on a coincident 
line, and this is true neither of the rigid circular arc nor of the 
rigid helix. 

In our pair of gears with infinitesimal teeth the line of contact 
must therefore be a straight line, and the working surfaces of the 
gears will be ruled surfaces or surfaces generated by the motions of 
a straight line. It follows, therefore, that if gears are to have line 
contact, their teeth must be built on ruled surfaces. 

These surfaces are called axodes or pitch surfaces, and we need 
only learn the capabilities of such surfaces to be able to predict all 
the possible forms of line contact gearing. 

96. A body may be located in a plane by locating two of its 
points. Let A,B and A',B' be two positions successively occupied 
by a body in a plane. A can move to A' by any one of an infinite 
number of circular arcs all lying in the plane, and the locus of their 
centers will be a straight line bisecting AA ' at right angles. Simi¬ 
larly with BB r . At the intersection of the two loci will be a point 
O, about which A can rotate to A' and B to B'. Hence any 
change of position in a plane may be produced by a simple rotation 
about a fixed point in the plane. An apparent exception to this is 
the particular case where the motion is one of simple translation. 
This need make no difficulty, however, the difference being merely 
that here the center O is at infinity. 

97. Applying the same reasoning to a spherical surface, we 
have the following : Any change of position in the surface of a 
sphere may be produced by a spheric rotation about a fixed point 
in the surface of the sphere. But any motion in the surface of a 
sphere is also a rotation about the center of the sphere ; it can, 
therefore, be produced by a simple rotation about a line joining the 
center with the point O in the surface. 

98. From the above, it follows that, “if a rigid solid body 
move in any way whatever, subject only to the condition that one 
of its points remains fixed, there is always one line of it through 
this point common to the body in any two positions.”* 

99. Consider, now, the most general motion of a rigid body of 
which no point is fixed. Let A and A' be one point of the body in 
any two positions. By a simple translatory motion, bring the point 
A' back to its first position A. The first and last positions have 
the point A in common. There is, therefore, some line passing 


* Euler’s theorem. See Thomson and Tait: Treatise on Natural Philosophy , Vol. I, p. 69. 

33 



through the point A , about which the body can be rotated from the 
first position to the last. During the translation, this line remained 
parallel to itself. Any change of position of a body can, therefore, 
be produced by a rotation and a translation. But the translation 
can be resolved into two components, one of which is parallel to the 
axis of rotation, and the other perpendicular thereto. The perpen¬ 
dicular translation and the rotation, being in the same plane, can 
be produced by a single rotation (§ 96). Therefore, any change of 
position of a body can be produced by one rotation about a fixed axis 
and one translation in the direction of the axis. 

100. Let any relative motion be resolved into an infinite num¬ 
ber of infinitesimal rotations and translations of the kind assumed 
above, and it follows that any motion can be produced by a com¬ 
bined rolling and sliding, the sliding being always in the direction 
of the instantaneous axis of rotation. 

Any motion can, therefore, be produced by the combined rolling 
and sliding of ruled surfaces having straight line contact. 

Hence any motion can be produced by a pair of line contact 
gears. Thus it is theoretically possible to attach a gear to each 
element of the cam shown in Fig. 44, so that for the motion allowed 
by the cam the gears will work together correctly and with line 
contact. 

101. Ruled surfaces are conveniently divided into four classes: 
the undevelopable warped surface, the developable warped surface, 
the cone, and the cylinder. The combinations which satisfy the 
requirements for pitch surfaces are eight in number. A pair of 
undevelopable warped surfaces may have rolling and sliding con¬ 
tact or rolling contact alone. The same is true of a pair of devel¬ 
opable warped surfaces and of a pair of cylinders. The two 
remaining combinations are : a pair of cones with rolling contact, 
and a cone and a developable warped surface with rolling and slid¬ 
ing contact. Each of the resulting motions can be cyclical ; that 
is, the gears can be brought back to a previous position without 
reversing. 

102. For most purposes gears are made to revolve about fixed 
axes with a constant velocity ratio. When this is the case the 
ruled surfaces must also be surfaces of revolution. Only five out 
of the eight combinations can be made to answer this condition. 

103. The undevelopable warped surfaces, which both roll and 
slide, can be hyperbolas of revolution, and, as such, they are the 


34 


true pitch surfaces of the skew gear, the worm gear, and, with one 
exception, of all forms of line contact gears having fixed axes that 
are neither parallel nor in the same plane. 

A particular case of the cone rolling and sliding on a developa¬ 
ble waiped surface is that of a right circular cone rolling and slid¬ 
ing on a circular disc. In this case the axes are neither parallel 
nor in the same plane ; but the velocity ratio must equal the sine of 
the angle between the axes, and the line of contact is in the supple¬ 
ment to the same angle. This form is best called the skew bevel 
gear, and is the exception referred to in the last paragraph. 

Two right circular cones rolling on each other are the pitch sur¬ 
faces of a pair of bevel gears, the axes of which will intersect 

Two right circular cylinders rolling, or rolling and sliding on 
each other, are the pitch surfaces of a pair of spur gears, the axes 
of which will be parallel. 

Observe that with a given pair of cylinders, the motion can be 
rolling or sliding or both, which is true of no other form. 



SKEW SKEW BEVEL BEVEL SPUR 


104. Fig. 45 shows the pitch surfaces of the four principal 
forms of line contact gears. 

FORMS OF GEAR TEETH. 

105. In the foregoing discussion the gear teeth were supposed 
to be of infinitesimal dimensions. It would be desirable to know 
what are the limitations as to form when the dimensions are finite ; 
but the general case involves some very complicated relations, and 
in a mathematical treatment there would be no fewer than six inde¬ 
pendent variables. The problem has never been solved, and it is 
doubtful whether a solution is possible. 


35 






































106. In practice little account is made of general theory ; but 
advantage is taken of a few particular cases that are simple enough 
to be easily understood and applied. 

Fig. 46 will give some idea of how 
complicated a tooth surface may be, 
and yet conform to the essential re¬ 
quirements. 

I07. The more difficult problems 
in gearing do not admit of graphical 
treatment ; for instead of plane curves there are undevelop¬ 
able surfaces to be dealt with. The easiest way to define such a 
surface as occurs in the perfect bevel gear, for example, is to con¬ 
struct the surface itself. When the surface is formed, as it fre¬ 
quently is by a machine designed especially for the purpose, there 
is evidently no need of a graphical treatment. The machine is 
made to cut the gear, and drawing instruments would only be in 
the way.* 



108. Our purpose being to learn the most direct methods of 
solving graphically such problems as are easiest solved graphically, 
our field narrows down to include only the spur gear and one or 
two approximations to other forms. 

109. The general condition that must be satisfied by the tooth 
of a spur gear appears to be as follows : 

Let Fig. 47 be a plane normal to the instantaneous axis at the 
point K. Such a plane will cut the pitch cylinders in two curves, 
AB and CD , called pitch lines. AB y CD , and EF , are three 
smooth curves which roll 
on each other in such 
manner that their com¬ 
mon tangent point is al¬ 
ways at K. Let the radii 
of curvature of the three 
curves at the point K be 
E, r, and p respectively, 
the direction in which the 
radius is measured being 



Fig*. 47 


indicated by prefixing the sign -f or—. Then ^-> -, and if 
this expression remains unchanged throughout the rolling, than 


* For information regarding the production of gears by machinery, see “Teeth 0/ Gears” 
by George B. Grant. 


'J 

a 


6 



any roulette GH generated by a point P of the curve EF rolling 
on the curve AB will be a possible tooth profile for the gear of 
which AB is the pitch line ; and the profile of the engaging tooth 
will be the roulette //, generated by the point P , as the curve EF 
rolls on the pitch line CD of the engaging gear. 



IIO. Fig. 48 will give a better idea of the import of the last 
paragraph. The full lines are possible tooth profiles for the pitch 
line AB , when rolling on the pitch line CD , while the dotted lines 
are impossible profiles. The largest possible tooth will have the 
profile GET, which is an epicycloid generated by rolling CD on AB. 



pinion. The form is arbitrary, but it satisfies the conditions of 
§ 109. It is desired to design a gear with 24 teeth that will prop¬ 
erly mesh with this pinion. Make the clearance and the back¬ 
lash -X" 


37 







112. Draw Fig. 49 by the dimensions given, leaving a space of 
8" above the tops of the teeth. To this figure is to be added the 
profiles of three teeth of the required gear. The pitch circles are 
to be placed tangent to each other, and the working faces of the 
teeth are to be placed in contact, so that the completed figure will 
represent the teeth in mesh. 

1 13. The back-lash is to be ytg -" measured on the pitch line. 
Let the curve A, Fig. 49, be revolved about the center of the pin¬ 
ion through an angle measured by y 1 ^" on the pitch line. It will 
fall in the position B. Draw B. This is a temporary alteration, 
and is to be made on only one of the teeth, as in Fig. 49. 

114. The envelope of the curve CBD will be the profile of a 
tooth of the required gear. It is found as follows : 

On a piece of sheet celluloid draw an arc of the pitch circle of 
the required gear (8" radius). Lay the celluloid over Fig. 49 (on 
the drawing board), placing the pitch circles tangent to each other 
at about E. Insert a needle at the point of tangency and trace the 
curve CBL >, or, more correctly, the curve FCBDG, for the points 
C and D must be distinct. . 

Rotate the celluloid about the needle as a pivot until the pitch 
lines appear to coincide for about D to the right of the needle, in¬ 
sert a second needle at the end of this coincidence (Y from the 
first), and, removing the first, turn the celluloid until the pitch lines 
appear to be tangent at the second needle point. Again trace the 
curve FCBDG , and so continue alternately rotating and tracing 
until the pitch lines are tangent near H. The envelope of the 

curves on the cellu¬ 
loid will be the en¬ 
velope required. 

115. Fig. 50 
shows the succes¬ 
sive positions of 
the pinion profile 
and their relation 
to the envelope. 

Observe that 
only a small part 
of each pinion pro¬ 
file is of use in determining the envelope. A little practice will 
enable one to save time by drawing only the necessary parts. 

38 












Lay the celluloid over Fig. 49, placing it in any one of the posi¬ 
tions it had during the process of getting the envelope, and transfer 
the envelope to the drawing board. The transferred curve will be a 
part of the profile of the required gear in proper mesh with the 
given pinion profile. 

This method of drawing gear teeth is known as the graphical 
moulding method. As described above, the only theoretical errors 
are due to the use of a finite number of instantaneous centers, and 
these errors are far within the errors of workmanship of the most 
accurate draughtsman or patternmaker. Use transparent sheet 
celluloid ".004 thick, with a dull finish on one side. It is much 
better than tracing paper. 

116. The gear teeth will be spaced 15 0 apart, there being 24 of 
them. The spacing can be done with the 6o° and 45 0 triangles, as 
in § 57 - 

Fig. 51 shows the two profiles in mesh. The envelope found by 
following the above directions will be the curve HIJKL. The 



rounded point J is cut off by a circular arc 8||" radius, and the parts 
H and L are replaced by an arbitrary curve N, such as will make 
the root of the tooth touch the circle M , 7^" radius. The clear¬ 
ance is thus made -fe". 

117. Besides Fig. 51, which should always be drawn full size, 
a patternmaker requires a sketch or drawing similar to Fig. 52, 


39 






which may conveniently be drawn to a smaller scale. The side 
view is really necessary only for wheels that have arms ; but the 
section cannot be omitted. 

118. Prob. 8. When one of the tooth profiles is a complete 
circle, the combination is known as a pin gear. 

The pin gear can be made to assume a variety of forms by 
changing the relative positions of the pins and the pitch circles. 



In Fig. 53, which represents the most common form, the contact 
occurs on only a small part of the circumference of each pin near 
the pitch line, and the conditions for correct tooth action (g 109), 
require that the center of the pin be placed inside of the pitch 
line A. 

119. The distance x, Fig. 54, at which the center must be placed 
within the pitch line A, is found as follows : CD is an epicycloid 
generated by rolling B on A , and CE is the evolute of CD. Then, 
of a pin F, whose circumference passes through C, the center must 


40 
















































































be on or within the curve CE , but not outside of the curve CE. It 
is possible to find an expression for the minimum value of x in 
terms of the diameters of the pin and the two pitch circles ; but 

this has never 
been done. 

In Fig. 53 the 
centers of the 
pins are placed 
inside of the 
pitch line A. 

12 0. The 
profile of the 
tooth that en¬ 
gages with the 
pin is parallel 
to the path of 
the center of 
the pin, and is 
found as fol¬ 
lows : 

Trace on celluloid an arc of the pitch line A, Fig. 53, and prick 
the position of an adjacent pin center. Roll the traced arc on the 
pitch line B, taking the instantaneous centers not more than \ n 
apart, and prick the successive positions of the pin center. This 
will give a series of points 
C in the path of the pin 
center. With a radius equal 
to the radius of the pins, 
draw circular arcs about 
the points C as centers, 
and the envelope of these 
arcs will be the tooth pro¬ 
file required. 

121 . The teeth found 
as above are made of such 
length outside of the pitch 
line that each one continues 
in contact until after the succeeding tooth begins contact, and the 
profiles within the pitch line B are made of any simple arbitrary 
form that will give a sufficient clearance. 



41 


















122. It may be observed that clock gear teeth are frequently 
given a form that is theoretically incorrect, for example, like that 
shown in Fig. 55, and sometimes the profile consists merely of a 
semicircular end tangent to two straight lines at the sides. We are 
thus led to the question, When are correct forms necessary ? 

123. Correctly formed teeth are of the greatest importance in 
gears that run at high speeds ; for at high speeds moving masses 
will, by virtue of their inertia, exert a considerable force in their 
endeavor to preserve a constant velocity ratio, and unless the tooth 
profiles have a corresponding shape, their contacts will consist of a 
series of blows instead of a steady pressure, the results being noise 
and rapid wear. For the same reason, the teeth must be equally 
spaced. The equal spacing is of even greater importance than the 
correct profiles. 

For heavily-loaded gears correct forms of teeth are desirable ; 
since their use will enable the pressure to be divided between two or 

three teeth, thus lessening the 
chances of breaking any one 
tooth. 

The function of clock 
gears is to transmit a mod¬ 
erate pressure from the main¬ 
spring to the escapement, and 
a motion from the escapement 
to the hands on the dial. In 
the transmission of the pres¬ 
sure a constant velocitv ratio is of no significance. In trans- 
mitting the motion, the small variations in the velocity ratio 
due to the incorrect forms of the teeth are unobjectionable, for 
the mean velocity ratio is determined absolutely by the num¬ 
bers of teeth. Next in importance is the equal spacing of the teeth, 
and least important of all is the correct form of the individual 
tooth. The clock is merely an instance in which the last condition 
need not be insisted upon, though there is no doubt but that a clock 
will work better and last longer by having correctly-formed gear 
teeth. 



124. A chamber wheel, Fig. 56, is a pair of gears inclosed in a 
fluid-tight case for the purpose of transforming a continuous rotary 
motion into a fluid displacement, or the reverse. They are used for 
pumps, blowers, engines, ventilators, water-meters, e\c. A great 


42 











many forms of the gear or rotating part of the chamber wheel have 
been devised ; but the principle of their action is essentially the 
same. The gears, by keeping in contact with each other, prevent 
the fluid from passing between them ; while each gear, by revolv¬ 
ing in contact with the outer casing, carries the fluid between its 
projecting teeth from the suction to the delivery side. The cham¬ 
ber gears are usually of such form that they are incapable of driv¬ 
ing each other, and so two ordinary spur gears are keyed to the 
shafts on the outside of the casing. 

125. “The theoretical volume of delivery for all forms of 
chamber wheels, whether continuous or intermittent in delivery, is 
practically equal to the volume described by the cross-section of a 
tooth of one of the two gears for each revolution.”* 

The great weakness of these wheels, which can never be quite 
overcome, is leakage ; and the superiority of any one form will be 
due to the facility with which machine tools can be made to cut 
with accuracy its contact surfaces. 

126. Prob. 9. Fig. 56 represents the form of chamber wheel 
known as the Root Blower. 

The two chamber gears, A,F, are cast from the same pattern. 
Their outer curved surfaces are finished ; also their ends, which are 
perfectly flat, and the short hubs projecting inwards from the ends 
are bored to 1^-" diam. for the shafts. The two semicylindrical shells 
C,D are cast from the same pattern, and finished by first planing the 
flat surfaces a, b, c, d. The two castings are then bolted together 
[a in contact with b, and c in contact with d ), and the insides of both 
are finished at one boring, and the flanges are faced. The two end 
plates, F,F, are cast from the same pattern; their flat surfaces, 
which come in contact with the ends of the chamber gears, are 
planed, and the holes are bored for the adjustable bushings, 
G,G,G,G. The end plates and the semicylindrical shells are fas¬ 
tened together by 24 bolts diam. Of the 24 bolts, at least 
eight must be “finished” or “fitted,” so that when the parts are 
fastened together the inner surfaces of the shells shall be exactly 
concentric with the bearings in the end plates. The remaining 16 
bolts may be left “rough ” or “black.” The upper shaft is length¬ 
ened at H for a driving pulley ; otherwise the two shafts are alike. 
The two gears, I,J , have cut teeth, and the blanks are cast from 
the same pattern. All parts are of cast .iron, except the shafts and 

*Reuleaux : The Constructor (Suplee’s trans.), p. 220. 


43 





bolts, which are of wrought iron, and the bushings, which are of 
brass. 

Besides the two views shown in Fig. 56, draw an end elevation 
at K , looking in the direction of the arrow, and leaving out one of 




_ 


CD- 



CO- 

CM- 



44 





























































































































































































the gear wheels, I,J. Also a full size cross-section of a chamber 

gear, as in Fig. 57, and detail views of the adjustable bushing, as 
in Fig. 58. 



127. In Fig. 57 the curves are all circular arcs, except that 
portion of the profile which is within the pitch circle. 

Draw Fig. 50, excepting the irregular curves, by the dimensions 
given, and then find the irregular profiles, as follows : 

Trace on a piece of celluloid the center line AB, and the pitch 
line Z, and prick the center JV. Place the celluloid so that the 
tracing of AB coin¬ 
cides with CB>, and 
the tracing of the 
pitch line is extern¬ 
ally tangent to Z. 

The pricked center 
will fall at n. From 
this position we are 
to roll the pitch line 
of the celluloid on 
the pitch line of the 
gear, and determine 
points in the path of 
n. The describing- 
point n is on the cellu¬ 
loid. Insert a needle 
at /, and around p as 
a center rotate the 
celluloid through a 
small angle. Insert 
a second needle at 
the point where the pitch circles intersect. Remove the first needle, 
and turn the celluloid about the second needle until the pitch 
circles are tangent at the second needle. Prick the position of n 
from the celluloid. By repeating the process any number of points 
can be found in the path of n. Observe, that in this case the pitch 
circles are of equal curvature, and hence the above method is 
theoretically correct, no matter how great the angle through which 
the celluloid is turned at each instantaneous center. 

The path of n will be a hypotrochoid, and the required profile is 
parallel to this path. Take the radius NA in the compasses, and, 


45 



































with centers at the various positions of n , draw the circular arcs R. 
The envelope of these arcs is the required profile. It will be 
seen that this chamber gear is very similar to the pin gear of Prob. 
8, the profile being parallel to the path of a point in both cases. 


128. T o 



take up the 
wear in a bear- 
i n g without 
disturbing the 
position of the 
center of the 


Fig*. 58 


FINISHED ALL OVER 


shaft, a special adjustable bushing, Fig. 58, is used. It consists of 
two sleeves screwed together by a buttress thread. The outside 
diameter of the thread is ij". The inner sleeve is slotted the 
whole length at A, and its flange is slotted at C and D t to make 
it elastic. The outer sleeve has six notches by which it can be 
turned with a spanner wrench. A dowel on the inside of the casing 
(AT, Fig. 56) engages one of the slots A,C, or Z>, Fig. 58, and pre¬ 
vents the bushing from turning with the shaft, at the same time 
allowing the bushing to be placed in any one of three positions to 
distribute the wear. A set screw, Fig. 56, holds the outer sleeve 
after it is tightened. 

Devise some way of continuously lubricating this bearing. 


46 


































\ 


1 









t 




































